Symmetry & critical points for a model shallow neural network
This provides insights into neural network training dynamics, but it is incremental as it builds on existing symmetry-based analyses.
The paper tackles the optimization problem of fitting two-layer ReLU networks by analyzing critical points using symmetry, showing that loss at spurious minima can decay to zero like k^{-1} or converge to a positive constant.
We consider the optimization problem associated with fitting two-layer ReLU networks with $k$ hidden neurons, where labels are assumed to be generated by a (teacher) neural network. We leverage the rich symmetry exhibited by such models to identify various families of critical points and express them as power series in $k^{-\frac{1}{2}}$. These expressions are then used to derive estimates for several related quantities which imply that not all spurious minima are alike. In particular, we show that while the loss function at certain types of spurious minima decays to zero like $k^{-1}$, in other cases the loss converges to a strictly positive constant. The methods used depend on symmetry, the geometry of group actions, bifurcation, and Artin's implicit function theorem.